To determine the length of the line segment [tex]\(\overline{WX}\)[/tex] with endpoints [tex]\(W(2, -7)\)[/tex] and [tex]\(X(5, -4)\)[/tex], we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] are the coordinates of point [tex]\(W\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of point [tex]\(X\)[/tex].
Given:
- [tex]\( W(2, -7) = (x_1, y_1) \)[/tex]
- [tex]\( X(5, -4) = (x_2, y_2) \)[/tex]
Substitute the coordinates into the distance formula:
[tex]\[ d = \sqrt{(5 - 2)^2 + (-4 + 7)^2} \][/tex]
Calculate each squared difference:
1. The difference in [tex]\( x\)[/tex] coordinates: [tex]\( 5 - 2 = 3 \)[/tex]
- Squaring this difference: [tex]\( 3^2 = 9 \)[/tex]
2. The difference in [tex]\( y \)[/tex] coordinates: [tex]\( -4 + 7 = 3 \)[/tex]
- Squaring this difference: [tex]\( 3^2 = 9 \)[/tex]
Add the squared differences:
[tex]\[ 9 + 9 = 18 \][/tex]
Take the square root of the sum:
[tex]\[ d = \sqrt{18} \][/tex]
Since [tex]\( \sqrt{18} \)[/tex] can be simplified further:
[tex]\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \][/tex]
Thus, the length of [tex]\(\overline{WX}\)[/tex] is:
[tex]\[ \boxed{3 \sqrt{2}} \][/tex]
Therefore, the correct answer is [tex]\( \boxed{E} \)[/tex].
